Let g be a probability density function, what is the ratio (g(x))/(g(t)) for x>t?

Let g be a probability density function, what is the ratio $\frac{g\left(x\right)}{g\left(t\right)}$ for $x>t$?
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Jovanni Salinas
Step 1
Continuity is important because we can always take a nice-behaved pdf like the Gaussian, and adjust it over a set of points with zero measure to get a PDF that doesn't strictly have a tail limit.
For example. Let $\varphi \left(x\right)$be the standard normal density function. It has nice tail properties and adheres to your conjecture.
Now, lets form the following function
$Q\left(x\right)=x{\mathbf{1}}_{x\in \mathbb{Z}}$
Step 2
Then the function $\psi \left(x\right)+Q\left(x\right)$ has not limit at either tail, yet it integrates to 1 and all that good stuff, since the domian where $Q\left(x\right)>0$ has zero Lebesgue measure.
So if g is a continuous pdf then limits should exist.